Cubic Graphs and Related Triangulations on Orientable Surfaces

摘要

Let $mathbb{S}_g$ be the orientable surface of genus $g$ for a fixed non-negative integer $g$.?We show that the number of vertex-labelled cubic multigraphs embeddable on $mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5/2(g-1)-1}gamma^{2n}(2n)!$, where $gamma$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally, for $gge1$, we prove that a typical cubic multigraph embeddable on $mathbb{S}_g$ has exactly one non-planar component.